Special Issue Article

Proc IMechE Part M:

J Engineering for the Maritime Environment

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Ó IMechE 2013

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DOI: 10.1177/1475090213502001

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Switching control of an underwaterglider with independently controllablewings

Andrea Caiti1, Vincenzo Calabro2, Stefano Geluardi3,

Sergio Grammatico4 and Andrea Munafo5

Abstract

The dynamic, control-oriented model of an underwater glider with independently controllable wings is presented. The

onboard vehicle’s actuators are a ballast tank and two hydrodynamic wings. A control strategy is proposed to improvethe vehicle’s maneuverability. In particular, a switching control law, together with a backstepping feedback scheme, is

designed to limit the energy-inefficient actions of the ballast tank and hence to enforce efficient maneuvers. The case

study considered here is an underwater vehicle with hydrodynamic wings behind its main hull. This unusual structure ismotivated by the recently introduced concept of the underwater wave glider, which is a vehicle capable of both surface

and underwater navigation. The proposed control strategy is validated via numerical simulations, in which the simulated

vehicle has to perform three-dimensional path-following maneuvers.

Keywords

Autonomous underwater vehicle, glider, wave glider, path-following

Date received: 18 March 2013; accepted: 22 July 2013

Introduction

Underwater gliders are winged autonomous under-

water vehicles (AUVs), which are usually employed for

long-term and flexible missions, unlike thrusters’ pro-

pelled underwater vehicles, such as oceanographic sam-

pling and underwater surveillance. We indeed remind

that the underwater glider concept1 motivated the

development of several prototypes, for instance,

‘‘Slocum,’’2‘‘Spray’’3 and ‘‘Seaglider.’’4 These vehicles

are all buoyancy-propelled fixed-winged gliders and

their attitude is controlled by shifting the internal bal-

last. See, for instance, the control-oriented approach

developed in Leonard and Graver.5

As underwater glider can be operated in high-

efficiency conditions of reliance on the battery power,

an accurate control system is crucial for their perfor-

mance and, especially, endurance. Most importantly,

the use of feedback control can provide robustness to

uncertainty and disturbances.5 Efficient state-feedback

control schemes are proposed in Bhatta and Leonard6

and Mahmoudian and Woolsey7 for underwater gliders

having fixed hydrodynamic wings.

More recently, the need for higher maneuverability

motivated the development of underwater gliders with

movable rudder8 or even with independently controlla-

ble wings.9 For instance, in Arima et al.,10 many experi-

mental tests are performed with the glider ‘‘Alex’’ in

order to characterize the so-called lateral response. This

latter characteristic is clearly not available if the glider

is equipped with hydrodynamic wings that are fixed.

This study is motivated by the recently introduced

concept of the underwater wave glider,11 capable of

both wave-propelled surface navigation and buoyancy-

driven underwater motions. The mentioned vehicle is

indeed a hybrid underwater vehicle, in the sense that it

is both a surface vehicle and an underwater glider. Its

structure is such that the hydrodynamic wings are

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1Department of Information Engineering, University of Pisa, Pisa, Italy2Cybernetics R&D, Kongsberg Maritime, Kongsberg, Norway3Max Planck Institute for Biological Cybernetics, Tubingen, Germany4Automatic Control Laboratory, ETH Zurich, Zurich, Switzerland5NATO Centre for Maritime Research and Experimentation, La Spezia,

Italy

Corresponding author:

Andrea Caiti, Department of Information Engineering, University of Pisa,

Pisa, Italy.

Email: a.caiti@dsea.unipi.it

PisPis

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positioned in the back of its main hull when the vehicle

operates as underwater glider.

Through this article, we present a three-dimensional

(3D) path-following problem,12,13 for our vehicle, and

from a control-oriented point of view, we propose a

state-feedback switching control scheme. This article is

organized as follows. Section ‘‘Underwater glider

model’’ presents the modeling of the vehicle and the

characterization of the actuators, namely, the ballast

and the independently controlled wings. Section

‘‘Underwater gliding control problem’’ presents a

switching control scheme for a path-following maneu-

ver. In order to validate the proposed control strategy,

many numerical simulations are performed and hence

presented in section ‘‘Simulations.’’ The last section

concludes the article.

Underwater glider model

In this section, we model the dynamics of the under-

water glider, with particular emphasis on the functional

characterization of the control actuators. The vehicle is

characterized by a variable mass m(t) and a center of

gravity (CoG) with variable position rg(t) due to the

(variable) quantity of water inside the ballast tank,

positioned on top of the vehicle’s hull.

For any generic vector v, we adopt the notation with

superscript ‘‘b’’ to indicate the components of v

expressed in the body-fixed frame ‘‘attached’’ to the

vehicle, while we use the superscript ‘‘n’’ to indicate the

navigation reference frame. According to Caiti et al.,11

the equations governing the dynamics of the position

and the velocity of the CoG are

rbg(t)=L+Y(t)

m(t), _rbg(t)=

_Y(t)

m(t)ÿ

_m(t)

m(t)rbg(t) ð1Þ

where L and Y(t) are, respectively, the static and the

dynamic contributes of the CoG dynamics.

Considering the generalized position h(t) 2 R6 (with

respect to an inertial frame), the generalized velocity

v(t) 2 R6 (with respect to the body-fixed frame) and the

mass of water e(t) 2 R contained in the ballast tank, a

13-dimensional model for the rigid body motion,

including the ‘‘ballast’s dynamics,’’ can be written as

follows

M(e) _v+C(v, e)v+D(v)v+ g(h, e)+T(v, e) _e= t ð2Þ

_h= J(h)v ð3Þ

_e= ue ð4Þ

where the mass matrix M(e) and the Coriolis matrix

C(e) already include the added mass effects,14D(v)v is

the hydrodynamic drag force acting on vehicle, g(h, e)

indicates the forces of gravity and buoyancy and the

term T(v, e) _e represents the effects of the variable CoG

(1). In equation (4), we assume that we can control the

velocity of the CoG _e via the control input ue. The gen-

eralized force t consists of the (controlled)

hydrodynamic effects of the wings, characterized in

detail later on. The matrix J(h) is the Jacobian matrix.

This modeling technique for vehicles having a vari-

able position of the CoG has been also used in Caiti

and Calabro15 for a hybrid AUV; see also Caffaz

et al.16 for further details. In Table 1, we show the

numerical parameters used in this work.

Ballast tank characterization

We assume that the position of the CoG can vary only

by filling and emptying the ballast tank. This means

that the position of the CoG, that is, rbg, can be written

as a static function of the mass of water e contained in

the ballast tank.

Based on the numerical parameters given in Table 1,

we provide a technical characterization of the ballast

actuator, positioned on top of the vehicle’s hull.

Therefore, injecting water in the ballast tank will result

in moving the CoG toward the top of the vehicle and

vice versa.

This choice also allows to regulate the pitch angle of

the vehicle by changing the water contained in the bal-

last tank (Figure 1). As a consequence, also the glide

angle of the vehicle is affected by the quantity of water

inside the ballast tank.

The buoyancy force of the vehicle depends linearly

on the amount of water embarked in the ballast tank.

We tune the position of the ballast tank so that the fol-

lowing symmetry is obtained for the vehicle’s buoy-

ancy: for values of water mass e. 0:5 kg, the vehicle

becomes ‘‘negative’’ (i.e. the gravity force is greater than

the buoyancy force), while for e\ 0:5 kg, the vehicle is

‘‘positive’’ (i.e. the gravity force is less than the buoy-

ancy force).

Table 1. Numerical parameters of the considered underwater

glider.

Characteristic glider parameters

VehicleLength 2 mDiameter 0.15 mStatic weight, ms 23.65 kgStatic CoG position, (L=ms)x 20.02 mStatic CoG position, (L=ms)y 0 mStatic CoG position, (L=ms)z 0.1 m

Ballast tankCapacity, max eb 1 kgMaximum rate, e_bjmax 0.1 kg/sPosition, (rbb)x 0.9 mPosition, (rbb)y 0 mPosition, (rbb)z 0.1 m

Hydrodynamic wingsWidth 0.25 mLength 0.10 mPosition, (rbw)x 20.8 mPosition, (rbw)y 60.2 mPosition, (rbw)z 0 m

CoG: center of gravity.

2 Proc IMechE Part M: J Engineering for the Maritime Environment 0(0)

As already mentioned before, by introducing a cer-

tain mass of water e in the ballast tank at the position

rb, we obtain a shift of the CoG rbg of the entire vehicle.

This is shown in the plot of Figure 2.

Wing characterization

The hydrodynamic wings play a fundamental role to

guarantee the gliding motion of the vehicle. Their con-

tribution affects the equations of the dynamics (2) via

the generalized force t.

For ‘‘numerical convenience,’’ throughout the arti-

cle, we consider the NACA009 wing profile. The orien-

tation of each wing depends on the angle of attack

(a+b) (see Figure 3), that is, the angle between the

longitudinal direction of the wing xv and the direction

of the velocity vv.

A hydrodynamic wing introduces the lift L(a+b)

and drag D(a+b) forces due to its relative velocity

with respect to the fluid velocity. A schematic

representation of the forces acting on the hydrody-

namic wing is given in Figure 3.

In this article, we consider the following approxima-

tions17 for such induced forces

L(a+b)=1

2rV2SCL(a+b) ð5Þ

D(a+b)=1

2rV2SCD(a+b) ð6Þ

where r is the water density, V is the wings’ speed in

the fluid, S is the area of wings’ surface, while the terms

CL and CD are dimensionless coefficients depending on

the chosen wings’ profile and also on the wings’ angle

of attack.

Underwater gliding control problem

We consider the gliding control problem, that is, we

look for a state-feedback control law ue(h, n, e) and an

appropriate wings’ orientation a, affecting the

AQ4AQ4Figure 1. Steady-state pitch angle reached by the vehicle as a function of the water contained in the ballast tank positioned on top

of the vehicle’s hull.

AQ5AQ5Figure 2. Variation in the longitudinal position of the CoG of the vehicle as a function of the water contained in the ballast tank

positioned on top of the vehicle’s hull.CoG: center of gravity.

Caiti et al. 3

generalized force as t= t(a), such that a certain

motion is achieved by the vehicle. We notice that set-

ting the wings’ orientation angle a to zero, we achieve

the same maneuverability of a glider with fixed wings.

Hence, we claim that this additional degree of freedom

can be exploited nontrivially, especially if the wings are

independently controlled.

We first notice what happens if the wings’ orienta-

tion angle a is fixed to zero. As shown in Figure 4, we

obtain that the equilibrium glide angle reached by the

vehicle is quite ‘‘small’’ if the amount of water con-

tained in the ballast tank is close to its central value,

which is 0.5 kg. On the contrary, the vehicle can achieve

larger gliding angles whenever we have values of water

mass e far from the central value.

We also remark that, in general, the control of the

wings positioned in the tail of the vehicle is not a trivial

task because the system suffers of ‘‘coupling effects.’’ In

fact, any control input on the wings introduces addi-

tional moments along the pitch axis, while the differen-

tial control strategy introduces moments along the

pitch and the roll axes.

Preliminaries on the path-following problem

We now define the path-following problem based on

the high-level path-following method presented in

Breivik and Fossen.18 The peculiarity of the method in

Breivik and Fossen18 is the development of control

laws for a generic ideal model of vehicle, according to

the following (backstepping) steps: (a) a point on the

vehicle (‘‘actual particle’’) tracks a desired trajectory

and (b) a control is constructed to achieve the desired

speed computed above.

This strategy is typical of path-following problems

as, unlike trajectory-tracking problems, no time limita-

tions are considered. The actual particle chosen is the

CoG, and the goal here is its convergence to the most

convenient ‘‘path particle,’’ also named ‘‘ideal particle.’’

Here, in our case study, the references are the pitch

and the yaw angles of the velocity of the ideal particle.

We refer and use the control scheme shown in Figure 5.

The model dynamics (2) are implemented in the block

Dynamics. The Reference block solves the step (a) of

the path-following problem described above, in the

sense that it computes the distance between the actual

particle and the ideal particle, together with the state

errors (~h, ~n, ~e). The Feedback block solves the step (b),

namely, it uses such values to generate the state-

feedback control actions.

Figure 3. Lift and drag forces (top) generated by the

hydrodynamic wings as a function of the angle of attack of the

hydrodynamic wings themselves (bottom). The ‘‘x-axis’’ of the

body-attached reference frame is usually not aligned, see the

angle b, with the direction ‘‘xw ’’ of the relative speed between

the hydrodynamic wing and the fluid.

AQ6AQ6Figure 4. Steady-state glide angle reached by the vehicle as a function of the water contained in the ballast tank positioned on top

of the vehicle’s hull.

4 Proc IMechE Part M: J Engineering for the Maritime Environment 0(0)

Switching state-feedback velocity control

It is well known that actuating the ballast tank at

‘‘high’’ frequency is particularly inefficient from the

energy consumption point of view. Namely, variations

in the quantity of water e in the tank should be used

only when really needed.

The purpose of the Feedback block is indeed to limit

the chattering behavior of the ballast actuator close to

the equilibrium point. This can be achieved by imple-

menting a switching control law, based on the ‘‘dis-

tance’’ from the equilibrium point.

Technically, we propose a feedback control of the

amount of water in the ballast tank that is purely pro-

portional far from the desired path, but null close to

the desired path. We can hence define the control law

ue(~h, ~n, ~e) :¼0:1 � sat(Ke

~e) if (~hT, ~nT)

4�e

0 otherwise

�

ð7Þ

where Ke 2 R is the feedback gain and �e. 0 is a certain

threshold to be tuned.

Namely, whenever the vehicle is ‘‘close enough’’ to

the desired path, the control of the ballast tank is dis-

abled to avoid the ballast chattering and only the wings

are used to finely reach the desired path.

Optimization-based wing control

According to the force characterization of the hydrody-

namic wings, previously shown in section ‘‘Wing char-

acterization,’’ the generalized force t applied to the

vehicle’s CoG is

t=B(b,a,V)a ð8Þ

where B(�) is a nonlinear function of the angles

b=(b1,b2)T, of the angles a=(a1,a2)

T and of the

wings’ velocity moduli (V1,V2)T =V 2 R2. The non-

linear function B(�) comes from the forces of lift L(�)and drag D(�), whose scheme is shown in Figure 3.

The objective here is indeed the choice of a to

recover the desired generalized force t, or a close

approximation of it, according to equation (8). For

simplicity, in the control design, we neglect the drag

terms because of about more than one order of magni-

tude smaller than the lift ones (in the desired operating

conditions). Therefore, we consider an approximate

nonlinearity

�t= �B(b,a,V)a ð9Þ

with �B(b,a,V) 2 R232 and �t=(t3, t6)T. However, we

remark that in the validation of the control law via

numerical simulations, although the control law is com-

puted neglecting the drag forces, all the terms are con-

sidered in the vehicle’s dynamics, including the drag

forces as well.

Regarding the control force �t, we choose these two

components, that is, t3 and t6 because the velocity

angles of pitch and yaw are used here as references. We

now present our switching control law, with the moti-

vation of achieving a ‘‘good’’ convergence to the refer-

ence path. In fact, we further remind that close to the

desired path, there is no control action due to the bal-

last tank in order to avoid an energy-inefficient actua-

tion. As shown later on via numerical simulations, this

design choice is particularly effective in this regard. We

indeed introduce an integral term ‘‘close’’ to the desired

path as follows

t�(~h, ~n, ~e) :¼Kp

~u~c

� �

if (~hT, ~nT)

4�e

Kp

~u~c

� �

+Ki

Ð ~u(t)~c(t)

� �

dt otherwise

8

>

>

<

>

>

:

ð10Þ

where Kp and Ki 2 R232 are the constant matrix gains

to be tuned.

A possible way to regulate the wing angles a is to

solve an optimization problem online. Given the cur-

rent angles a0 andb0, the current velocity moduli V0

and a desired force t� to be reproduced, the angles a

can be chosen as follows

a :¼ argminx2R2

s(x)TQs(x)+ (xÿ a0)TR(xÿ a0)

subject to : a4x4�a, Da4xÿ a04Da

s(x) :¼ t� ÿ �B(b0, x,V0)x

ð11Þ

The bounds a, �a, Da,Da are free design parameters

reflecting the physical limits on the wing actuators. The

weight matrices Q,R are positive definite.

The meaning of the optimization problem (11) is that

we choose an admissible angle a, which is sufficiently

close to its present value a0, namely, to avoid chattering

on the wings, and also such that the error s between the

induced force and the desired one is sufficiently small.

However, we immediately notice that problem (11) is a

nonlinear optimization problem as s(�) is a nonlinear

function.

Following the approach proposed in Johansen

et al.,19 we can consider a linearized version of problem

Figure 5. Backstepping control scheme. The reference block

computes the reference signals and hence the state errors. The

feedback block implements the control laws to track the

reference signals.

Caiti et al. 5

(11), obtaining the following quadratic problem (QP)

to be solved online

a :¼argminx2R2

�s(x)TQ�s(x)+ (xÿ a0)TR(xÿ a0)

subject to : a4x4�a, Da4xÿ a04Da

�s(x) :¼ t� ÿ �B b0,a0,V0ð Þa0

ÿ∂

∂a�B(b,a,V)að Þ

� �

(b0,a0,V0)

� (xÿ a0)

ð12Þ

Note that, unlike Johansen et al.,19 the approximation

of the original nonlinearity B(�) in equation (8) with�B(�) in equation (9) is such that �B(b,a,V) is never sin-

gular. Within this simplifying reduction, there is no

need to address the problem via sequential QPs.

Simulations

We first have performed many numerical simulations

with the goal of tuning the free design parameters intro-

duced in the previous sections. Then, we validate our

proposed switching control strategy. From our numeri-

cal experience, we have noticed that in our backstep-

ping control scheme, an undesired chattering behavior

of the ballast actuator would be generated if merely lin-

ear control laws are employed.

Our simulation experience indeed confirms the fol-

lowing fact. The action of the ballast tank is more rele-

vant to regulate the pitch variable, while the corrections

determined by the independently controlled hydrody-

namic wings play a relevant role in regulating the yaw

variable. Basically, a differential control of the hydro-

dynamic wings allows to generate roll and yaw motions.

The numerical parameters used in the implemented QP

are a,a=6158, Da,Da=658, Q= I and R=20I.

We present here the gliding motion of the glider, as

shown in Figure 6, that has to follow an helical

trajectory of radius 40 m (see Figure 6 (right)). Such

trajectories are particularly efficient to explore and

sample underwater environments, for instance, sink

holes.20,21 The initial conditions of the vehicle are

h0 =(0, 70, ÿ 20, 0, 0, 0)T and v0 =(0:05, 0, 0, 0, 0, 0)T.The initial position error vector is chosen as

~h0 =(0, 30, ÿ20, 0, 0, 0)T. The initial value of the water

contained in the ballast tank is eb =0:5. The controllergain has been tuned as

Ke =1, Kp =ÿ6 0

0 3

� �

, Ki =ÿ0:02 0

0 0:01

� �

Turning off the ballast tank actuator, when the vehi-

cle is ‘‘close enough’’ to the path, contributes to having

low-energy consumption by correcting trajectory only

through the wings’ regulation. Furthermore, the intro-

duction of the integral term helps to track the desired

path with small state error as shown in Figure 7.

Figure 7 also shows the angular wings’ actuation and

the amount of water in the ballast tank. Figure 7(e)

and (f) shows the angular errors ~u and ~c.

We now show the numerical results obtained if the

control system does not switch to the local control law.

Figure 8 clearly shows that when the vehicles get quite

close to the desired path, the ballast actuation starts

chattering, causing an undesired behavior for the whole

controlled vehicle. We indeed heuristically claim that

switching to a local control law improves the closed-

loop performance of the vehicle when following a

desired path.

Simulations in the presence of a constant oceanic

current

We now simulate the case in which there is a constant

oceanic current perturbing the dynamics of the

Figure 6. Three-dimensional plot of the vehicle tracking the (a) desired trajectory and (b) desired helical trajectory and actual

trajectory of a characteristic point of the vehicle.

6 Proc IMechE Part M: J Engineering for the Maritime Environment 0(0)

underwater vehicle. We hence consider a constant force

perturbation tp :¼ (0:1, 0, 0:2, 0, 0, 0)T, whose modulus

is about one-tenth of the one of the hydrodynamic

forces.

In order to obtain good closed-loop performances,

we ‘‘increment’’ the control gains as follows

Ke =1, Kp =ÿ8 0

0 4

� �

, Ki =ÿ0:04 0

0 0:02

� �

Figure 9 shows the numerical results of the simula-

tions. The controlled vehicle still performs in a satisfy-

ing way, which shows some kind of robustness of the

proposed control algorithm.

AQ7AQ7Figure 7. (a) Time evolution of the position error. Time evolution of the actuator signals: (b) water contained in the ballast tank,

(c) left-wing angle and (d) right-wing angle. Time evolution of the angular errors: (e) pitch and (f) yaw errors.

Caiti et al. 7

Conclusion

We have presented a control-oriented model of an

underwater glider with independently controllable

hydrodynamic wings. Through the use of a ballast tank

and the hydrodynamic wings, a simple, backstepping,

control scheme has been implemented to accomplish

some particular 3D path-following maneuvers. We

have used a switching control scheme in order to avoid

the undesired chattering of the ballast tank actuator

when the vehicle gets close to the desired path. The con-

trol algorithm is validated via numerical simulations.

We remark that our particular choice of a switching

control law is only justified by simulation heuristics,

not by theoretical arguments.

Figure 8. Linear feedback control with no switching. (a) Time evolution of the position error. Time evolution of the actuator

signals: (b) water contained in the ballast tank, (c) left-wing angle and (d) right-wing angle. Time evolution of the angular errors: (e)

pitch and (f) yaw errors.

8 Proc IMechE Part M: J Engineering for the Maritime Environment 0(0)

However, the provided example shows the potential-

ities of using independent actuations for the hydrody-

namic wings. In particular, the actuation of the ballast

tank can be limited by just exploiting the corrections of

the wings. This would further increase the energy effi-

ciency of underwater gliders, even if nontrivial maneu-

vers are to be performed.

Therefore, we expect that this work can motivate

further research studies on the design of more accurate

control schemes exploiting the independent actuation

of the hydrodynamic wings, besides the physical reali-

zation of glider prototypes of the kind described here.

From this latter point of view, we finally remark that

the realization of a prototype capable of exploiting its

Figure 9. The presence of a perturbing oceanic current. (a) Time evolution of the position error. Time evolution of the actuator

signals: (b) water contained in the ballast tank, (c) left-wing angle and (d) right-wing angle. Time evolution of the angular errors:

(e) pitch and (f) yaw errors.

Caiti et al. 9

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hydrodynamic wings together with the waves’ energy

for the surface navigation, and together with the ocea-

nic currents for the underwater navigation, would be

an important breakthrough for the oceanic engineering

community.

Declaration of conflicting interests

The authors declare that there is no conflict of interest.

Funding

This research received no specific grant from any fund-

ing agency in the public, commercial or not-for-profit

sectors.

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